# nLab horizontal categorification

Contents

category theory

## Applications

#### Algebra

higher algebra

universal algebra

categorification

# Contents

## Idea

Horizontal categorification or Oidification describes the process by which

1. a concept is realized to be equivalent to a certain type of category or magmoid with a single object;

2. and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.

## Remarks

• This is to be contrasted with vertical categorification.

• It can be argued that the term ‘categorification’ should be reserved for vertical categorification, since we can use ‘oidification’ for the horizontal concept.

• It has rightly been remarked that groupoids are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.

## Examples

• The horizontal categorification of groups are groupoids: categories in which every morphism is invertible.

• A horizontal categorification of algebras are algebroids: enriched categories in the category of vector spaces.

• A horizontal categorification of rings are ringoids: enriched categories over the category of abelian groups. (blog)

• A horizontal categorification of $C^*$-algebras hence ought to be known as $C^*$–algebroids but is usually known as C*-categories.

• Since, by the Gelfand-Naimark theorem, C-star algebras are dual to topological spaces, Paolo Bertozzini et. al proposed to define spaceoids to be entities dual to $C^*$-categories (blog).

• And finally the exception to the rule: a many-object monoid is not called a monoidoid – but is called a category!

On the other hand, if the category is $R$-linear (enriched in R Mod) then it may again be referred to as an algebroid over $R$.

## Further discussion

Related $n$-Café-discussion is in

Last revised on August 12, 2022 at 07:30:15. See the history of this page for a list of all contributions to it.